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We consider a system of n coupled KPZ equations on \T=[0,1), coupled through the quadratic nonlinearities. The coupling constant \gamma^i_{j,k} which multiplies \nabla (h^j h^k) in the ith equation, together form an n\times n\times n real tensor \Gamma = \{\gamma^i_{j,k}\}_{1\leq i,j,k\leq n}. When \Gamma is trilinear, that is, when the values of \gamma^i_{j,k} are the same under permutations of i,j,k, the system has a global-in-time solution starting from, say, a Wiener invariant measure.
There is debate about the associated lateral and transversal scalings of the solutions, which would indicate their fluctuation class. In this talk, after some background, we discuss necessary/sufficient conditions on \Gamma, found via invariant theory, so that the system decouples under an orthogonal transformation \sigma \in O(n), that is when (\sigma h)_i solves independent 1D KPZ equations. These are explicit when n \leq 3. In such a case, the fluctuation class can be identified by those of the 1D marginal systems, well-known as either KPZ or EW classes. We also discuss the situation of `partial' decoupleability when, under a \sigma, at least one of the equations splits off from the others. This is based on a recent arXiv paper with B. Fu, T. Funaki, and S.C. Venkataramani.
Coupled KPZ equations and their decoupleability
